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Series: Dissertation Defense

Series: Dissertation Defense

In this thesis, we extend De Giorgi's interpolation method to a class of parabolic equations which are not gradient flows but possess an entropy functional and an underlying Lagrangian. The new fact in the study is that not only the
Lagrangian may depend on spatial variables, but also it does not induce a metric. Assuming the initial condition is a density function, not necessarily smooth, but solely of bounded first moments and finite entropy, we use a variational scheme to
discretize the equation in time and construct approximate solutions. Moreover, De
Giorgi's interpolation method reveals to be a powerful tool for proving convergence
of our algorithm. Finally, we analyze uniqueness and stability of our solution in L^1.

Series: Dissertation Defense

This disseratation provides the conceptual development, modeling and simulation, physical implementation and measured hardware results for a procticable digital coherent chaotic communication system.

Series: Dissertation Defense

Series: Dissertation Defense

Series: Dissertation Defense

Series: Dissertation Defense

Series: Dissertation Defense

The first part of this work deals with open dynamical systems. A natural question of how the survival probability depends upon a position of a hole was seemingly never addresses in the theory of open dynamical systems. We found that this dependency could be very essential. The main results are related to the holes with equal sizes (measure) in the phase space of strongly chaotic maps. Take in each hole a periodic point of minimal period. Then the faster escape occurs through the hole where this minimal period assumes its maximal value. The results are valid for all finite times (starting with the minimal period), which is unusual in dynamical systems theory where typically statements are asymptotic when time tends to infinity. It seems obvious that the bigger the hole is the bigger is the escape through that hole. Our results demonstrate that generally it is not true, and that specific features of the dynamics may play a role comparable to the size of the hole.
In the second part we consider some classes of cellular automata called Deterministic Walks in Random Environments on \mathbb Z^1. At first we deal with the system with constant rigidity and Markovian distribution of scatterers on \mathbb Z^1. It is shown that these systems have essentially the same properties as DWRE on \mathbb Z^1 with constant rigidity and independently distributed scatterers. Lastly, we consider a system with non-constant rigidity (so called process of aging) and independent distribution of scatterers. Asymptotic laws for the dynamics of perturbations propagating in such environments with aging are obtained.